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Don't put words in my keyboard. The only one who believes in remote dilation as a frame-independent phenomenon here is you! That remote dilation is only in your mind. You don't even understand that, which says all about you as a "thinker." The fact of whether something suffers frame-dependent dilation or not does involve having things other than photons in it. In fact, in order to write down the geodesic equation for photons you must do an affine transformation, because their proper time is identically zero, so no common-sense clocking will help you describe their histories. But what am I telling you; you know next to nothing about relativity, that not being the worst. The worst being that you don't bother to examine your own assumptions, or anybody's criticism. Can you imagine Einstein telling Hilbert "please consider my silly mistake as a valid assumption"? Einstein quickly re-wrote his paper. Learnt much from Hilbert, and went on to publish one of the most important papers in the history of physics.

I will pop up every now and then to see what experts and other serious thinkers have to say. Your post is only valuable in that sense. The only trouble is I will have to check for you twisting everything I or anybody else has said, just because you don't understand the first thing about relativity, you don't read what you write, let alone others, and you stubbornly stick to a bunch of silly pseudo-scientific propositions to no end.

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Don't put words in my keyboard. The only one who believes in remote dilation as a frame-independent phenomenon here is you! That remote dilation is only in your mind. You don't even understand that, which says all about you as a "thinker." The fact of whether something suffers frame-dependent dilation or not does involve having things other than photons in it. In fact, in order to write down the geodesic equation for photons you must do an affine transformation, because their proper time is identically zero, so no common-sense clocking will help you describe their histories. But what am I telling you; you know next to nothing about relativity, that not being the worst. The worst being that you don't bother to examine your own assumptions, or anybody's criticism. Can you imagine Einstein telling Hilbert "please consider my silly mistake as a valid assumption"? Einstein quickly re-wrote his paper. Learnt much from Hilbert, and went on to publish one of the most important papers in the history of physics.

I will pop up every now and then to see what experts and other serious thinkers have to say. Your post is only valuable in that sense. The only trouble is I will have to check for you twisting everything I or anybody else has said, just because you don't understand the first thing about relativity, you don't read what you write, let alone others, and you stubbornly stick to a bunch of silly pseudo-scientific propositions to no end.

If I've mischaracterized your position on this then I apologize. I guess I don't know what your position is, at all.

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If it "arises from the metric" for specified circumstances then why can't we calculate a global time dilation field for that same solution and given conditions?

Because - as I have attempted to explain - time dilation is a relationship between distant clocks, whereas a field assigns a particular object (a tensor of spinor of any rank) to each local event in spacetime. You cannot point to an event in spacetime and say “I am going to assign time dilation factor X to this event”, without any further qualification - this does not make any physical sense.

The most fundamental entity in GR (and the solution to the Einstein field equations) is the metric tensor field - it assigns a metric tensor to each event in spacetime. To put it in the simplest possible terms, the metric tensor field allows you to quantify how each event in spacetime is related to all other events - both in spatial terms, and in terms of time. It does so by defining a mathematically precise relationship between neighbouring events, so that, by integrating along curves, you can calculate relationships between more distant events, e.g. the length of a world line connecting them.

Time dilation in GR is a geometric property of world lines, in that it is the ratio between the lengths of world lines between the same events - the total time a clock accumulates between two given events is equivalent to the geometric length of the world line traced out by that clock. And how long that world line will be depends on the geometry of the spacetime it is in, and what kind of world line it is.

Take for example a rotating spherical body, such as a planet. If you let a test clock orbit the planet once in its direction of rotation, starting and finishing at some point P, then that orbit will take a total time T1. If you now start at the same spot P, but orbit in the opposite direction (counter the planet’s direction of rotation, but along the same orbit, with all other initial and boundary conditions remaining equal), you will get some orbital time T2, which will be ever so slightly different. That’s because, even though you start at the same point P, and traverse the same spatial distance along the same orbit, the geometry of spacetime is such that the lengths of the two world lines will differ. The ratio between these two geometric lengths is one example of gravitational time dilation - the value of that ratio depends on where the point P is, the initial and boundary conditions of the clock kinematics, and the global geometry of the underlying spacetime. How would you capture all this by assigning a single value to point P, as you seem to want to do with your “time dilation field” idea?

Again, on closer consideration, in order to capture all relevant degrees of freedom so that all aspects of gravity can be correctly modelled, independently of the precise circumstances, at least a rank-2 tensor field is necessary. That’s what GR does.

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Because - as I have attempted to explain - time dilation is a relationship between distant clocks, whereas a field assigns a particular object (a tensor of spinor of any rank) to each local event in spacetime. You cannot point to an event in spacetime and say “I am going to assign time dilation factor X to this event”, without any further qualification - this does not make any physical sense.

Well I understood that rather impressive explanation. +1

Only one cooment.
Marcus is talking 'field' in the mathematical sense, not the Physical sense. They can be slightly different.

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Because - as I have attempted to explain - time dilation is a relationship between distant clocks, whereas a field assigns a particular object (a tensor of spinor of any rank) to each local event in spacetime. You cannot point to an event in spacetime and say “I am going to assign time dilation factor X to this event”, without any further qualification - this does not make any physical sense.

[...]

Boy, was that a good explanation! +1

14 hours ago, joigus said:

That's only because I'm a remote object, and you're giving me the wrong coordinates.

And this was intended as a joke. Coordinates mean nothing, it's the metric tensor contracted with the coordinates, as Markus so brilliantly has explained. Just to clarify...

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How would you capture all this by assigning a single value to point P, as you seem to want to do with your “time dilation field” idea?

Your example is well described, thank-you. The proper time of any object would still depend on its path through the field, so the point P wouldn't be expected to hold all relevant information. Regardless, if clock A went East and clock B went West they would still move through equivalent scalar fields of equal length. I'll have to think about it. I suspect it's related to the gradient of the field.

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OK I've given this more thought. I think until we can come up with an analytic proof (or disproof) we are left trying to find a scenario where gravity "exists" but time dilation does not (OR time dilation exists but gravity does not). Frame-dragging around a rotating object does cause both gravity and time dilation effects depending on direction of orbit.

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We are most certainly not; maybe you are, and have been for some time.
( may be time to look for a new hobby )
The rest of us are satisfied with the GR model, which, other than a few areas of applicability, works extremely well.

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Fair point, *I* will work on an analytic proof. In the mean time, if Markus can think of a scenario where gravity and time dilation are independent of one another in GR I would be very interested to learn about it.

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It's a good thought, but I believe in this very thread we've had links showing that linear momentum is a source of gravitational attraction.

Not quite.

You need to study the Principle of equivalence in regards to the time dilation of moving bodies.

The shorthand for the principle is inertial mass is equivalent to gravitational mass.

[math] m_i=m_g[/math]

Relativistic moving bodies gain inertial mass (used to be called relativistic mass) however the current accepted term is the variant mass. The rest mass is now described as in invariant mass. (Mass all observers can agree on). The variant mass obviously depends on the observer.

6 hours ago, rjbeery said:

OK I've given this more thought. I think until we can come up with an analytic proof (or disproof) we are left trying to find a scenario where gravity "exists" but time dilation does not (OR time dilation exists but gravity does not). Frame-dragging around a rotating object does cause both gravity and time dilation effects depending on direction of orbit.

If you have any form of mass including gravitational mass then you will always have time dilation. However miniscule.

To give you a clear example time dilation has been measured at a single meter difference in elevation.

PS glad to see your using the term fields as well as starting to look more into current studies of relativity. +1.

This is an open source textbook on SR that the author wrote to cover many of the misconceptions common to science forums. The author is a PH.D who specializes in SR and GR. I have had numerous conversations with him over the last decade or so.

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OK I've given this more thought. I think until we can come up with an analytic proof (or disproof) we are left trying to find a scenario where gravity "exists" but time dilation does not (OR time dilation exists but gravity does not). Frame-dragging around a rotating object does cause both gravity and time dilation effects depending on direction of orbit.

Well, you can consider a hollow sphere made from a thin shell of matter. The exterior of the shell looks like any other spherically symmetric body, so it is described by the usual Schwarzschild metric. The hollow interior of the shell however is a different story - no tidal gravity is detected therein, meaning a test particle placed anywhere into the interior remains at rest. At the same time though, if you place a clock into the interior, and somehow compare its tick rate against a reference clock far away on the outside, you will find that it is time dilated, even though no forces (which would cause it to move) are detected locally where the clock is. So this would be an example of time dilation, but no tidal forces. Another even simpler example would be a uniformly accelerated frame in an otherwise empty region of spacetime; again, an accelerated clock is dilated, but there is no tidal gravity.

A real-world example of a case where you have tidal gravity in the spatial part of the metric, but no time dilation, would be a region of spacetime that is uniformly filled with dust, in a way that ensures homogeneity and isotropy. The FLRW metric - on which our current understanding of cosmology, the Lambda-CDM model, is based - is an example of this. In this metric the temporal part is constant, but the spatial part is not.

I should also mention here that within metrics, each coordinate coefficient can depend on all coordinates, including time. So not only can things vary as you move in space, they can also vary with time, and with any possible combination of the two. So you can get quite complicated spacetimes that are neither static nor stationary, with highly non-intuitive geometries. And if that wasn’t enough, then it needs mentioning that the dynamics of GR are highly non-linear, meaning gravity self-interacts; hence (at least in principle) you can have topological constructs that are formed and held together purely by their own gravitational self-energy, in the complete absence of any “traditional” sources.

I think you are beginning to see now that the dynamics of spacetime are very rich and varied - they can’t be captured by just assigning some scalar field. I actually seem to remember having once seen a formal proof that a rank-2 tensor is the lowest rank object required to capture all dynamics of GR, I just can’t remember where I have seen it. If I come across it, I will post it here.

I actually seem to remember having once seen a formal proof that a rank-2 tensor is the lowest rank object required to capture all dynamics of GR, I just can’t remember where I have seen it. If I come across it, I will post it here.

I too recall such a proof, if I can remember the source I will post it.

Edited May 22 by Mordred

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I actually seem to remember having once seen a formal proof that a rank-2 tensor is the lowest rank object required to capture all dynamics of GR, I just can’t remember where I have seen it. If I come across it, I will post it here.

3 hours ago, Mordred said:

I too recall such a proof, if I can remember the source I will post it.

Interesting... Tidal effects come to mind at your suggestion, because those are second-order effects, which requires an order-2 tensor.

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Well, you can consider a hollow sphere made from a thin shell of matter. The exterior of the shell looks like any other spherically symmetric body, so it is described by the usual Schwarzschild metric. The hollow interior of the shell however is a different story - no tidal gravity is detected therein, meaning a test particle placed anywhere into the interior remains at rest.

Ahh yes, I'm familiar with the shell theorem. I misspoke when I said "find a scenario where gravity "exists" but time dilation does not (OR time dilation exists but gravity does not)" because I've already mentioned that it's the gradient that determines gravitational attraction (in this model). Maybe a better way to say it is that I need to "find a scenario where the time dilation gradient and gravitational attraction differ"?

10 hours ago, Markus Hanke said:

A real-world example of a case where you have tidal gravity in the spatial part of the metric, but no time dilation, would be a region of spacetime that is uniformly filled with dust, in a way that ensures homogeneity and isotropy. The FLRW metric - on which our current understanding of cosmology, the Lambda-CDM model, is based - is an example of this. In this metric the temporal part is constant, but the spatial part is not.

I don't understand this description but I'll look into it. I can't picture in my mind's eye how a cloud of dust would create tidal gravity but no time dilation. A volume of dust has mass, and mass causes time dilation.

10 hours ago, Markus Hanke said:

I think you are beginning to see now that the dynamics of spacetime are very rich and varied - they can’t be captured by just assigning some scalar field. I actually seem to remember having once seen a formal proof that a rank-2 tensor is the lowest rank object required to capture all dynamics of GR, I just can’t remember where I have seen it. If I come across it, I will post it here.

I would be very interested in seeing this. How certain are we that there are no redundancies in GR? As an off-the-cuff example, we know of the mass-energy equivalence but we still account for mass separately in the equations.

In the end, obviously, no amount of abstract discussion is going to sway any opinions. If and when I discover more analytic proof I'll share it here.

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I believe that in accordance with the third law of Hegel's dialectics ( development goes in a spiral), physics will somehow return to the ether as a medium for the propagation of interactions. Of course, not to the Lorentz model, but on a new level. Most likely, the term "ether" itself will be replaced with another one.However, this is already happening, because the vacuum in the modern sense is not a void.

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If you have a trivial counter-example, please share it. If you don't, then how do you draw the above conclusion?

Two clocks in relative motion will have symmetric time dilation values, based only on their relative speed. One clock has a different mass than the other, so they do not have the same momentum. It has to be a separate effect.

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Maybe a better way to say it is that I need to "find a scenario where the time dilation gradient and gravitational attraction differ"?

I am honestly not sure if I follow your thought process correctly, since such a notion as “time dilation gradient” does not make much sense to me. But nonetheless, the aforementioned case of an orbit around a rotating mass should be an example. Another scenario that immediately comes to mind would be two parallel beams of light (or any other pp-wave spacetime, for that matter) - if you fire two parallel beams of light in the same direction, there will be no gravitational attraction between them, even though they carry energy. But if you fire the same two beams of light so that they are initially parallel, but travel in opposite directions (i.e. you let emitter and receiver trade places for one of the beams), then they will indeed experience a gravitational attraction.

18 hours ago, rjbeery said:

Maybe a better way to say it is that I need to "find a scenario where the time dilation gradient and gravitational attraction differ"?

There will of course be time dilation between a clock inside the volume of dust, and some other reference clock outside of the dust cloud; but there is no time dilation between two clocks that are both located inside the dust cloud. I should have been more clear on this, as I was initially thinking of the cosmological case, where there is no “outside”.

18 hours ago, rjbeery said:

I would be very interested in seeing this.

As I said, I don’t immediately recall where I saw that proof, it was a few years back when I came across it, and it was in a printed textbook. However, I can offer an outline (!) of my own attempt at proving this, for whatever it is worth.

For this, allow me to go back to the basics, and consider what it actually means for a manifold (such as spacetime) to have curvature, and how to capture this mathematically. Imagine you choose some arbitrary point P on your manifold, and pick out an arbitrary tangent vector attached to that point. Now you parallel-transport that tangent vector around a small (i.e. infinitesimal) loop that starts and ends at your point P. The question is - will the initial vector before the parallel transport operation coincide with the final vector at the end of the procedure, regardless of the specific curve the loop describes, and what direction I travel on that loop? On a flat manifold, using standard calculus, the answer is obviously yes (I use single bars “|” to denote ordinary derivatives), since ordinary derivatives commute:

\[A_{\mu |\nu \gamma } -A_{\mu |\gamma \nu } =0\]

However, if we allow the manifold to not be flat, then the situation changes; following the standard prescription for this (refer to any textbook on differential geometry), we must now replace ordinary with covariant derivatives, which do not in general commute. The degree to which they fail to commute is (I use double bars “||” to denote covariant derivatives):

The object \(R_{\mu \nu \gamma \delta}\) is called the Riemann curvature tensor, and it uniquely specifies all aspects of the geometry of a given manifold. The question then becomes how you explicitly calculate the components of the Riemann tensor, i.e. what kind of object is it a function of? For this you need to only remember that GR uses the Levi-Civita connection, which is torsion free; this implies symmetry in the lower indices of the Christoffel symbols:

This being the case, you can then work out an explicit coordinate expression for the Riemann tensor from the above equations. I won’t typeset it here now since it is tedious to write in LaTeX notation (you can easily Google it, if you are interested) - I will simply point out that the Riemann tensor turns out to be a function of the connection coefficients and their derivatives only, which in turn are functions of the metric tensor and its derivatives only.

So in other words, and that is the point of this whole exercise, given the fact that GR uses the Levi-Civita connection to describe parallel transport, a unique description of all relevant aspects of a manifold’s geometry under GR (i.e. the Riemann curvature tensor) arises from a rank-2 tensor, being the metric tensor. A simple accounting of the indices in the above expressions show that there is no mathematical possibility of any lower rank object (such as a scalar or vector) doing the same job. Which is what we wanted to show.

The above is obviously only an outline - you could fill in the details and actual calculations yourself, using any standard textbook on differential geometry. I don’t know how rigorous the above really is, but that’s how I would approach such a proof - quantify the failure of derivatives to commute on curved manifolds; then, given a connection, check what kind of object the coordinate expression for the curvature tensor depends on. It seems pretty simple and logical to me. But if someone here who is actually an expert in the area can think of a better, more rigorous way, or can point out an error in the above reasoning, then I would definitely be interested in seeing it!

Edited May 23 by Markus Hanke

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A couple of points to add to the excellent post above. A scalar is rank 0. A vector is rank 1. A vector has both magnitude and direction. You need a higher rank when you require two vectors. Such as the example given by Markus.

If I recall correctly the Kronecker delta function is also rank two. If I'm correct then hermitean groups would also be rank 2 but that's just a side note.

The Poincare group is SO(3.1) which GR falls under. Which is a double cover [math]SU(2)\otimes SU(2)/\mathbb{Z}^2[/math] .

So even in tensor ranks you require a minimal rank 2. Just to provide a tensor example. (Each of those groups is a tensor. The SO(3.1) is a 4×4 while each SU(2) is a 2×2 The Z parity operator is also 2×2.

The proofs I have come across on rank 2 requirement were tensor related proofs. Which I looking for a more understandable example as they tend to be too complex for the average poster.

Edit I did a quick search and I am correct the Kronecker delta function is a rank two tensor.

It will also be a valuable tool to better understand rotations of the tensors.

Such an example of tensor rotations is when you must rotate the Minkowskii or Lorentz tensor to describe acceleration (rapidity requires a rotation) or boost (A boost is also a type of rotation).

Brian Crowell gave examples of each in that SR textbook I previously linked and provides some greater detail.

A little side note the best tool to master GR is to study differential geometry. Once you understand differential geometry for Euclidean and curved geometries understanding GR8 becomes incredibly easy. (You won't even require Tensors ) they are a tool to handle multiple unknowns in essence an organization tool to keep track of multiple unknowns)

Edited May 23 by Mordred

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As I said, I don’t immediately recall where I saw that proof, it was a few years back when I came across it, and it was in a printed textbook. However, I can offer an outline (!) of my own attempt at proving this, for whatever it is worth.

Leading up to Marcus' comment I have marked the relevant result on the third page.

@rjbeery I don't know how deeply you want to go but this is a simple derivation with one static source of gravitational potential. Your theory must predict/reproduce what happens with multiple sources.

I don't know if anyone has considered time dilation at lagrange points and lines where there are at least two such potentials.